Finite Group Databasebeta

← C4×#456×C127 #44585145

Subgroups — C4×#456×C127 #44585145

Order 32512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×2−1+4×C127C211yesmaximal, characteristic
2C2×#456×C127C211yesmaximal, characteristic
3C4×#137×C127C212yesmaximal, complemented
4C4×C4⋊Q8×C127C211yesmaximal, characteristic, complemented
5C127×#1886C211yesmaximal, characteristic, complemented
6C127×#1882C212yesmaximal, complemented
7C127×#1859C212yesmaximal, complemented
8C127×#1857C211yesmaximal, characteristic, complemented
9C127×#1793C211yesmaximal, characteristic
10C4×#104×C127C212yesmaximal, complemented
11C4×#101×C127C211yesmaximal, characteristic, complemented
12C2×2−1+4×C127C2211yescharacteristic
13C2×#137×C127C2212yes—
14C2×C4⋊Q8×C127C2211yescharacteristic
15#520×C127—22no—
16#520×C127—22no—
17#520×C127C2212yes—
18#519×C127—22no—
19#519×C127C2211yescharacteristic
20C4×#41×C127—22no—
21C4×#41×C127—22no—
22C4×#41×C127C2212yes—
23C2×C4×Q8×C127—22no—
24C2×C4×Q8×C127C2211yescharacteristic
25C2×C4×Q8×C127C2211yescharacteristic
26#456×C127C418yesdirect product factor
27#441×C127—22no—
28#440×C127—22no—
29C4×Q16×C127—22no—
30C4×SD16×C127—22no—
31#428×C127C2212yes—
32C2×#104×C127C2212yes—
33C2×#104×C127C2212yes—
34#419×C127C2212yes—
35#417×C127C2212yes—
36C2×#101×C127C2211yescharacteristic
37C2×#101×C127C2211yescharacteristic
38C4×M16×C127C2212yes—
39#391×C127C2211yescharacteristic
40#391×C127C2211yescharacteristic
41#389×C127—22no—
42#384×C127C2211yescharacteristic
43C4×C4⋊C4×C127—22no—
44C43×C127C2211yescharacteristic
45#343×C127C2212yes—
46#339×C127C2212yes—
47#339×C127C2212yes—
482−1+4×C127—24no—
492−1+4×C127—22no—
502−1+4×C127C2×C414yes—
51C2×#41×C127—22no—
52C2×#41×C127—22no—
53C2×#41×C127C2312yes—
54C22×Q8×C127—22no—
55C22×Q8×C127C2311yescharacteristic
56C22×Q8×C127C2311yescharacteristic
57#137×C127—24no—
58#137×C127C2×C418yescomplemented
59C2×Q16×C127—22no—
60C2×SD16×C127—22no—
61C2×M16×C127C2312yes—
62C2×M16×C127C2312yes—
63C4⋊Q8×C127—22no—
64C4⋊Q8×C127C2×C414yescomplemented
65C4×Q8×C127—22no—
66C4×Q8×C127—24no—
67C4×Q8×C127—22no—
68C4×Q8×C127—22no—
69C4×Q8×C127—22no—
70C4×Q8×C127—22no—
71C4×Q8×C127—22no—
72C4×Q8×C127—22no—
73C4×Q8×C127—22no—
74C4×Q8×C127—22no—
75C4×Q8×C127D412yes—
76C4×Q8×C127D412yes—
77C4×D4×C127—22no—
78C4×D4×C127—24no—
79C4×D4×C127—22no—
80C4×D4×C127—22no—
81C4×D4×C127—22no—
82C4×D4×C127—24no—
83C4×D4×C127—22no—
84C4×D4×C127D412yes—
85C4×D4×C127D412yes—
86#117×C127—24no—
87#117×C127—24no—
88#117×C127—22no—
89#117×C127—22no—
90#117×C127—22no—
91#117×C127C2312yes—
92C2×C4⋊C4×C127—24no—
93C2×C4⋊C4×C127—22no—
94C2×C4⋊C4×C127—22no—
95C2×C4⋊C4×C127—21no—
96C2×C4⋊C4×C127—22no—
97C2×C4⋊C4×C127D412yes—
98C2×C4⋊C4×C127C2311yescharacteristic
99C2×C4⋊C4×C127C2311yescharacteristic
100C2×C42×C127—24no—
101C2×C42×C127—21no—
102C2×C42×C127—22no—
103C2×C42×C127—21no—
104C2×C42×C127—21no—
105C2×C42×C127D412yes—
106C2×C42×C127C2311yescharacteristic
107C2×C42×C127C2311yescharacteristic
108C2×C42×C127C2311yescharacteristic
109#107×C127—22no—
110#106×C127—22no—
111#104×C127—24no—
112#104×C127C2×C418yescomplemented
113#103×C127—22no—
114#103×C127—22no—
115#103×C127—22no—
116#102×C127—22no—
117#101×C127—22no—
118#101×C127C2×C414yescomplemented
119#97×C127—22no—
120C4×C8×C127—22no—
121#95×C127—22no—
122#95×C127—22no—
123#41×C127—28no—
124#41×C127—28no—
125#41×C127—28no—
126#41×C127—28no—
127#41×C127—22no—
128#41×C127—22no—
129#41×C127C22×C414yes—
130#41×C127#4114yes—
131C2×Q8×C127—22no—
132C2×Q8×C127—28no—
133C2×Q8×C127—28no—
134C2×Q8×C127—22no—
135C2×Q8×C127—22no—
136C2×Q8×C127—21no—
137C2×Q8×C127—22no—
138C2×Q8×C127—22no—
139C2×Q8×C127C22×C412yes—
140C2×Q8×C127C22×C412yes—
141C2×Q8×C127C2×D412yes—
142C2×Q8×C127#4112yes—
143C2×D4×C127—22no—
144C2×D4×C127—22no—
145C2×D4×C127C2×D412yes—
146C22×C4×C127—24no—
147C22×C4×C127—21no—
148C22×C4×C127—22no—
149C22×C4×C127—21no—
150C22×C4×C127—21no—
151C22×C4×C127—24no—
152C22×C4×C127—22no—
153C22×C4×C127—22no—
154C22×C4×C127—21no—
155C22×C4×C127C2×D412yes—
156C22×C4×C127C2×D411yescharacteristic
157C22×C4×C127C2×D412yes—
158C22×C4×C127C2×D411yescharacteristic
159C22×C4×C127C2411yescharacteristic
160Q16×C127—28no—
161SD16×C127—28no—
162M16×C127—22no—
163M16×C127—22no—
164M16×C127—22no—
165M16×C127C22×C414yes—
166C2×C8×C127—22no—
167C2×C8×C127—22no—
168C4⋊C4×C127—24no—
169C4⋊C4×C127—22no—
170C4⋊C4×C127—22no—
171C4⋊C4×C127—28no—
172C4⋊C4×C127—22no—
173C4⋊C4×C127—24no—
174C4⋊C4×C127—22no—
175C4⋊C4×C127—22no—
176C4⋊C4×C127—22no—
177C4⋊C4×C127—44no—
178C4⋊C4×C127—42no—
179C4⋊C4×C127C2×D412yes—
180C4⋊C4×C127C2×D412yes—
181C22⋊C4×C127—24no—
182C22⋊C4×C127—22no—
183C22⋊C4×C127—22no—
184C22⋊C4×C127—22no—
185C42×C127—24no—
186C42×C127—24no—
187C42×C127—24no—
188C42×C127—22no—
189C42×C127—22no—
190C42×C127—22no—
191C42×C127—22no—
192C42×C127—22no—
193C42×C127—21no—
194C42×C127—22no—
195C42×C127—42no—
196C42×C127—41no—
197C42×C127—41no—
198C42×C127C22×C412yes—
199C42×C127C2×D412yes—
200C42×C127C2×D412yes—
201C23×C127—22no—
202C23×C127C22×D411yescharacteristic
203Q8×C127—28no—
204Q8×C127—24no—
205Q8×C127—22no—
206Q8×C127—24no—
207Q8×C127—24no—
208Q8×C127—42no—
209Q8×C127C4×D414yes—
210D4×C127—28no—
211D4×C127—24no—
212D4×C127—24no—
213D4×C127—22no—
214D4×C127C4×D414yes—
215C2×C4×C127—22no—
216C2×C4×C127—22no—
217C2×C4×C127—216no—
218C2×C4×C127—22no—
219C2×C4×C127—22no—
220C2×C4×C127—22no—
221C2×C4×C127—22no—
222C2×C4×C127—24no—
223C2×C4×C127—24no—
224C2×C4×C127—22no—
225C2×C4×C127—22no—
226C2×C4×C127—22no—
227C2×C4×C127—22no—
228C2×C4×C127—22no—
229C2×C4×C127—22no—
230C2×C4×C127—41no—
231C2×C4×C127—41no—
232C2×C4×C127—41no—
233C2×C4×C127—41no—
234C2×C4×C127—41no—
235C2×C4×C127—44no—
236C2×C4×C127—42no—
237C2×C4×C127—42no—
238C2×C4×C127—41no—
239C2×C4×C127—82no—
240C2×C4×C127—84no—
241C2×C4×C127C4×D414yes—
242C2×C4×C127#12011yescharacteristic
243C2×C4×C127#12012yes—
244C2×C4×C127#12011yescharacteristic
245C2×C4×C127C23×C411yescharacteristic
246C2×C4×C127C22×D412yes—
247C2×C4×C127C2×#4112yes—
248C2×C4×C127C2×#4111yescharacteristic
249C8×C127—24no—
250C8×C127—42no—
251C22×C127—21no—
252C22×C127—21no—
253C22×C127—24no—
254C22×C127—42no—
255C22×C127—42no—
256C22×C127C2×C4×D411yescharacteristic
257C22×C127C2×#12011yescharacteristic
258C22×C127#52511yescharacteristic
259C4×C127—24no—
260C4×C127—21no—
261C4×C127—22no—
262C4×C127—24no—
263C4×C127—24no—
264C4×C127—42no—
265C4×C127—42no—
266C4×C127—42no—
267C4×C127—42no—
268C4×C127—42no—
269C4×C127—81no—
270C4×C127#45612yesdirect product factor
271C4×C127C2×C4×D412yes—
272C4×C127#52512yes—
273C4×#456C12711yesmaximal, characteristic, direct product factor, Sylow
274C2×C127—21no—
275C2×C127—21no—
276C2×C127—44no—
277C2×C127C4×#12011yescharacteristic
278C2×C127C2×#45611yescharacteristic
279C2×C127#301811yescharacteristic
280C4×2−1+4C2×C12711yescharacteristic
281C2×#456C2×C12711yescharacteristic
282C4×#137C2×C12712yescomplemented
283C4×C4⋊Q8C2×C12711yescharacteristic, complemented
284#1886C2×C12711yescharacteristic, complemented
285#1882C2×C12712yescomplemented
286#1859C2×C12712yescomplemented
287#1857C2×C12711yescharacteristic, complemented
288#1793C2×C12711yescharacteristic
289C4×#104C2×C12712yescomplemented
290C4×#101C2×C12711yescharacteristic, complemented
291C127C4×#45611yescharacteristic, direct product factor, Sylow
292C2×2−1+4C22×C12711yescharacteristic
293C2×#137C22×C12712yes—
294C2×C4⋊Q8C22×C12711yescharacteristic
295#520—22no—
296#520—22no—
297#520C22×C12712yes—
298#519—22no—
299#519C22×C12711yescharacteristic
300C4×#41—22no—
301C4×#41—22no—
302C4×#41C22×C12712yes—
303C2×C4×Q8—22no—
304C2×C4×Q8C22×C12711yescharacteristic
305C2×C4×Q8C22×C12711yescharacteristic
306#456C4×C12718yesdirect product factor
307#441—22no—
308#440—22no—
309C4×Q16—22no—
310C4×SD16—22no—
311#428C22×C12712yes—
312C2×#104C22×C12712yes—
313C2×#104C22×C12712yes—
314#419C22×C12712yes—
315#417C22×C12712yes—
316C2×#101C22×C12711yescharacteristic
317C2×#101C22×C12711yescharacteristic
318C4×M16C22×C12712yes—
319#391C22×C12711yescharacteristic
320#391C22×C12711yescharacteristic
321#389—22no—
322#384C22×C12711yescharacteristic
323C4×C4⋊C4—22no—
324C43C22×C12711yescharacteristic
325#343C22×C12712yes—
326#339C22×C12712yes—
327#339C22×C12712yes—
3282−1+4—24no—
3292−1+4—22no—
3302−1+4C2×C4×C12714yes—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41C23×C12712yes—
334C22×Q8—22no—
335C22×Q8C23×C12711yescharacteristic
336C22×Q8C23×C12711yescharacteristic
337#137—24no—
338#137C2×C4×C12718yescomplemented
339C2×Q16—22no—
340C2×SD16—22no—
341C2×M16C23×C12712yes—
342C2×M16C23×C12712yes—
343C4⋊Q8—22no—
344C4⋊Q8C2×C4×C12714yescomplemented
345C4×Q8—22no—
346C4×Q8—24no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×Q8D4×C12712yes—
356C4×Q8D4×C12712yes—
357C4×D4—22no—
358C4×D4—24no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—24no—
363C4×D4—22no—
364C4×D4D4×C12712yes—
365C4×D4D4×C12712yes—
366#117—24no—
367#117—24no—
368#117—22no—
369#117—22no—
370#117—22no—
371#117C23×C12712yes—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4D4×C12712yes—
378C2×C4⋊C4C23×C12711yescharacteristic
379C2×C4⋊C4C23×C12711yescharacteristic
380C2×C42—24no—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42D4×C12712yes—
386C2×C42C23×C12711yescharacteristic
387C2×C42C23×C12711yescharacteristic
388C2×C42C23×C12711yescharacteristic
389#107—22no—
390#106—22no—
391#104—24no—
392#104C2×C4×C12718yescomplemented
393#103—22no—
394#103—22no—
395#103—22no—
396#102—22no—
397#101—22no—
398#101C2×C4×C12714yescomplemented
399#97—22no—
400C4×C8—22no—
401#95—22no—
402#95—22no—
403#41—28no—
404#41—28no—
405#41—28no—
406#41—28no—
407#41—22no—
408#41—22no—
409#41C22×C4×C12714yes—
410#41#41×C12714yes—
411C2×Q8—22no—
412C2×Q8—28no—
413C2×Q8—28no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8C22×C4×C12712yes—
420C2×Q8C22×C4×C12712yes—
421C2×Q8C2×D4×C12712yes—
422C2×Q8#41×C12712yes—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4C2×D4×C12712yes—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4C2×D4×C12712yes—
436C22×C4C2×D4×C12711yescharacteristic
437C22×C4C2×D4×C12712yes—
438C22×C4C2×D4×C12711yescharacteristic
439C22×C4C24×C12711yescharacteristic
440Q16—28no—
441SD16—28no—
442M16—22no—
443M16—22no—
444M16—22no—
445M16C22×C4×C12714yes—
446C2×C8—22no—
447C2×C8—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—28no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4C2×D4×C12712yes—
460C4⋊C4C2×D4×C12712yes—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C42—24no—
466C42—24no—
467C42—24no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—21no—
474C42—22no—
475C42—42no—
476C42—41no—
477C42—41no—
478C42C22×C4×C12712yes—
479C42C2×D4×C12712yes—
480C42C2×D4×C12712yes—
481C23—22no—
482C23C22×D4×C12711yescharacteristic
483Q8—28no—
484Q8—24no—
485Q8—22no—
486Q8—24no—
487Q8—24no—
488Q8—42no—
489Q8C4×D4×C12714yes—
490D4—28no—
491D4—24no—
492D4—24no—
493D4—22no—
494D4C4×D4×C12714yes—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—216no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4—84no—
521C2×C4C4×D4×C12714yes—
522C2×C4#120×C12711yescharacteristic
523C2×C4#120×C12712yes—
524C2×C4#120×C12711yescharacteristic
525C2×C4C23×C4×C12711yescharacteristic
526C2×C4C22×D4×C12712yes—
527C2×C4C2×#41×C12712yes—
528C2×C4C2×#41×C12711yescharacteristic
529C8—24no—
530C8—42no—
531C22—21no—
532C22—21no—
533C22—24no—
534C22—42no—
535C22—42no—
536C22C2×C4×D4×C12711yescharacteristic
537C22C2×#120×C12711yescharacteristic
538C22#525×C12711yescharacteristic
539C4—24no—
540C4—21no—
541C4—22no—
542C4—24no—
543C4—24no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—81no—
550C4#456×C12712yesdirect product factor
551C4C2×C4×D4×C12712yes—
552C4#525×C12712yes—
553C2—21no—
554C2—21no—
555C2—44no—
556C2C4×#120×C12711yescharacteristic
557C2C2×#456×C12711yescharacteristic
558C2C127×#301811yescharacteristic
subgroupG/H
centerC2×C4×C127#120
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC22×C127C2×#120
prime radicalC2×2−1+4×C127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C2×C42×C127 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C4×#137×C127C2semidirect
4. C4×C4⋊Q8×C127C2semidirect
5. C127×#1886C2semidirect
6. C127×#1882C2semidirect
7. C127×#1859C2semidirect
8. C127×#1857C2semidirect
10. C4×#104×C127C2semidirect
11. C4×#101×C127C2semidirect
26. #456×C127C4direct (N × H)
58. #137×C127C2×C4semidirect
64. C4⋊Q8×C127C2×C4semidirect
112. #104×C127C2×C4semidirect
118. #101×C127C2×C4semidirect
270. C4×C127#456direct (N × H)
273. C4×#456C127direct (N × H)
282. C4×#137C2×C127semidirect
283. C4×C4⋊Q8C2×C127semidirect
284. #1886C2×C127semidirect
285. #1882C2×C127semidirect
286. #1859C2×C127semidirect
287. #1857C2×C127semidirect
289. C4×#104C2×C127semidirect
290. C4×#101C2×C127semidirect
291. C127C4×#456direct (N × H)
306. #456C4×C127direct (N × H)
338. #137C2×C4×C127semidirect
344. C4⋊Q8C2×C4×C127semidirect
392. #104C2×C4×C127semidirect
398. #101C2×C4×C127semidirect
550. C4#456×C127direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×2−1+4×C1272
C2×#456×C1272
C4×#137×C1272
C4×C4⋊Q8×C1272
C127×#18862
C127×#18822
C127×#18592
C127×#18572
C127×#17932
C4×#104×C1272
C4×#101×C1272
C4×#456127